Exploring Albert Rijksbaron's book, The Syntax and Semantics of the Verb in Classical Greek: An Introduction, to see how it would need to be adapted for Koine Greek. Much of the focus will be on finding Koine examples to illustrate the same points Rijksbaron illustrates with Classical examples, and places where Koine Greek diverges from Classical Greek.

RandallButh wrote:I think it's mainly an extension of verbs that like the perfect naturally.

e.g., εἴσεται can be called future perfect, but it is really just a future to οἶδεν.
τεθνηξεται (I wouldn't expect this word with the απο prefix in future perfect but haven't checked)
μεμνησεται

The future of ἕστηκα / εἱστήκει would be (irregularly from this present pattern under discussion) be σταθήσεται, right?

No, σταθήσεται is future (middle-)passive.

Smyth wrote:§584. Future Perfect Active.—The future perfect active of most verbs is formed periphrastically (600). Two perfects with a present meaning, ἕστηκα I stand (ἵστημι set) and τέθνηκα I am dead (θνῄσκω), form the future perfects ἑστήξω I shall stand, τεθνήξω I shall be dead.

Smyth wrote:§1958. When the perfect has the force of a present, the future perfect is used like a simple future (1946): κεκλήσομαι I shall bear the name, μεμνήσομαι shall remember, κεκτήσομαι shall possess. So in the two active forms: τεθνήξω I shall be dead, ἑστήξω I shall stand.
a. The aorist subjunctive with ἄν (2324), not the future perfect, is used to denote a past action in relation to an action still in the future.

I remember my surprise at seeing both those future perfect forms of θνῄσκειν and ἴστασθαι in a single line in Aeschylus; I know it was in the Oresteia but I can't remember exactly where it was.